A general geometric construction of coordinates in a convex simplicial polytope

Barycentric coordinates are a fundamental concept in computer graphics and geometric modeling. We extend the geometric construction of Floater's mean value coordinates [Floater, M.S., Kós, G., Reimers, M., 2005. Mean value coordinates in 3d. Computer Aided Geometric Design 22 (7) (2005) 623–631; Ju, T., Schaefer, S., Warren, J., 2005a. Mean value coordinates for closed triangular meshes. In: Proceedings of ACM SIGGRAPH 2005] to a general form that is capable of constructing a family of coordinates in a convex 2D polygon, 3D triangular polyhedron, or a higher-dimensional simplicial polytope. This family unifies previously known coordinates, including Wachspress coordinates, mean value coordinates and discrete harmonic coordinates, in a simple geometric framework. Using the construction, we are able to create a new set of coordinates in 3D and higher dimensions and study its relation with known coordinates. We show that our general construction is complete, that is, the resulting family includes all possible coordinates in any convex simplicial polytope.